G-equivariant φ-coordinated quasi modules forquantum vertex algebras
Haisheng Li1
Department of Mathematical SciencesRutgers University, Camden, NJ 08102, USA
Abstract
This is a paper in a series to study quantum vertex algebras and theirrelations with various quantum algebras. In this paper, we introduce a notionof T-type quantum vertex algebra and a notion of G-equivariant φ-coordinatedquasi module for a T -type quantum vertex algebra with an automorphismgroup G. We refine and extend several previous results and we obtain acommutator formula for G-equivariant φ-coordinated quasi modules. As anillustrating example, we study a special case of the deformed Virasoro algebraVirp,q with q = −1, to which we associate a Clifford vertex superalgebra andits G-equivariant φ-coordinated quasi modules.
1 Introduction
Previously, inspired by Etingof-Kazhdan’s theory of quantum vertex operator al-gebras (see [EK]), we developed a theory of (weak) quantum vertex algebras andtheir modules in [Li2] and [Li3]. It was shown (see [Li3], [KL]) that quantum vertexalgebras in this sense can be associated to Zamolodchikov-Faddeev algebras of a cer-tain type. To associate more (quantum) algebras including quantum affine algebraswith quantum vertex algebras, we furthermore developed in [Li6] a theory of whatwe called φ-coordinated quasi modules for a weak quantum vertex algebra. Indeed,this new theory enables us to associate quantum affine algebras with weak quantumvertex algebras conceptually. It then remains to realize this conceptual associationexplicitly and to study the associated quantum vertex algebras and their modules,including their φ-coordinated quasi modules.
In this paper, we develop the theory of φ-coordinated quasi modules furtherand we refine and extend some of the results therein, paving the way to explicitlyassociate quantum vertex algebras to quantum affine algebras. Specifically, we in-troduce a notion of T -type quantum vertex algebra and a notion of G-equivariantφ-coordinated quasi module for a T -type quantum vertex algebra V , where G isan automorphism group of V equipped with a linear character χ. Among themain results of this paper, we establish a commutator formula for G-equivariantφ-coordinated quasi modules for a T-type weak quantum vertex algebra. As anillustrating example, we apply this theory to the deformed Virasoro algebra Virp,qwith q = −1 and we obtain a natural connection of this algebra with a concreteClifford vertex superalgebra.
1Partially supported by NSA grant H98230-11-1-0161 and China NSF grant (No. 11128103)
1
Now, we give a more detailed account of this paper. First, the notion of quantumvertex algebra formulated in [Li2] was somewhat derived from Etingof-Kazhdan’snotion of quantum vertex operator algebra. Quantum vertex algebras in this senseare generalizations of vertex algebras and vertex super-algebras, whereas quantumvertex operator algebras in the sense of Etingof-Kazhdan are formal deformationsof vertex algebras. Among the main ingredients for a quantum vertex algebra V isa rational quantum Yang-Baxter operator S(x) which governs a generalized com-mutativity (namely locality) of the associated vertex operators on V . On the onehand, this notion singles out of a large family of vertex algebra-like structures, andon the other hand, this theory is not much more complicated than the ordinaryvertex algebra theory.
In regard to the theory of φ-coordinated quasi modules for weak quantum vertexalgebras, this very parameter φ is what was called an associate of the one-dimensionaladditive formal group (law) F (x, y) = x + y, where an associate is a formal seriesφ(x, z) ∈ C((x))[[z]], satisfying
φ(x, 0) = x and φ(x, φ(y, z)) = φ(x+ y, z) (= φ(F (x, y), z)). (1.1)
It was shown that every associate (of the additive formal group law) can be explicitly
constructed by φ(x, z) = ezp(x)d
dx ·x with p(x) ∈ C((x)). In particular, taking p(x) = 1one gets φ(x, z) = x + z = F (x, z) (the formal group itself) and taking p(x) = xone gets φ(x, z) = xez. The truth is that the ordinary theory of (weak) quantumvertex algebras and modules is governed by the formal group law, whereas to eachassociate φ one can attach a theory of φ-coordinated quasi modules. In the theoryof weak quantum vertex algebras and modules, an important role was played by theassociativity
Y (u, x0 + x2)Y (v, x2) = Y (Y (u, x0)v, x2) (1.2)
(unrigorous), which is commonly referred to as operator product expansion by physi-cists. In contrast, in the theory of φ-coordinated (quasi) modules, this associativityis replaced with
Y (u, φ(x2, x0))Y (v, x2) = Y (Y (u, x0)v, x2) (1.3)
(unrigorous). For most of the better known algebras including quantum affine al-gebras, we take φ(x, z) = xez. It was proved therein that for any highest weightmodule W for a quantum affine algebra of a fixed level, the Drinfeld generatingfunctions, viewed as fields on W , generate a weak quantum vertex algebra in a cer-tain natural way with W as a canonical φ-coordinated quasi module. In this way,quantum affine algebras were tied to weak quantum vertex algebras conceptually.
Recall that a rational quantum Yang-Baxter operator is an important ingredientfor a quantum vertex algebra. In this paper, we define a T-type quantum vertex alge-bra to be a quantum vertex algebra V such that the rational quantum Yang-Baxteroperator S(x) is associated to a trigonometric quantum Yang-Baxter operator T (x)
2
with S(x) = T (ex). Let V be a T-type quantum vertex algebra and let G be anautomorphism group equipped with a linear character χ : G→ C×. We then definea notion of G-equivariant φ-coordinated quasi V -module. In the definition, in ad-dition to the associativity (1.3) with φ(x2, x0) = x2e
x0 , two defining properties arethat
YW (gv, x) = YW (v, χ(g)x) for g ∈ G, v ∈ Vand that for any u, v ∈ V , there exists a polynomial p(x) with only zeroes in{χ(g) | g ∈ G} such that
p(x/z)YW (u, x)YW (v, z) ∈ Hom(W,W ((x, z))).To a certain extent, this notion of G-equivariant φ-coordinated quasi V -module
is similar to the notion of twisted V -module (see [FLM], [FFR], [D]) for a vertexalgebra V . Note that a twisted V -module is associated to an automorphism σ offinite order N with a choice of an N -th root of unity, which amounts to choosing alinear character for 〈σ〉 = ZN , and the twisted vertex operator map satisfies a similarG-equivariance property. For a vertex operator algebra V , a notion of ZN -covariantquasi module was introduced and a canonical isomorphism between the category ofZN -twisted modules and the category of ZN -covariant quasi modules was establishedin [Li5]. As one of the main results of this paper, we obtain a commutator formula,which is somewhat analogous to that for twisted modules.
In this paper, as an illustrating example we apply this theory to the deformedVirasoro algebra Virp,q with q = −1. The deformed Virasoro algebra Virp,q withnonzero complex parameters p and q was introduced and studied in [SKAO], [AKSO],and the special case with q = −1 was specially studied in [BP]. The deformed Vira-soro algebra is among the simplest quantum algebras in the sense that it is generatedby one single field. We here establish a canonical isomorphism between the categoryof highest weight modules for Virp,q with q = −1 and the category of G-equivariantφ-coordinated quasi modules for a specific Clifford vertex super-algebra.
As for quantum affine algebras Uq(ĝ) (with q a nonzero complex number), ourspeculation is that for every � ∈ C, we shall have a canonical T-type quantum vertexalgebra with an automorphism group G isomorphic to a subgroup of C× such thathighest weight modules for Uq(ĝ) of level � are exactly G-equivariant φ-coordinatedquasi modules for the conjectured T -type quantum vertex algebra.
In a pioneer work [FR], E. Frenkel and N. Reshetikhin introduced a theory ofdeformed chiral algebras. Among the key ingredients in the notion of deformedchiral algebra are two spaces, called the space of states and the space of fields, anda trigonometric quantum Yang-Baxter operator on the space of fields. This theoryand the Etingof-Kazhdan theory look quite different and they have been studiedindependently. Now, the structure of a deformed chiral algebra can be enhancedto a T -type quantum vertex algebra V equipped with an automorphism group Gtogether with a G-equivariant φ-coordinated quasi module.
This paper is organized as follows: Sections 2 and 3 are preliminary; In Section2 we present some basic results on formal calculus, and in Section 3 we recall the
3
basic notions and results about (weak) quantum vertex algebras. In Section 4, we de-fine the notions of T -type quantum vertex algebra and G-equivariant φ-coordinatedquasi modules, and we present several results including the commutator formula. InSection 5, we study the deformed Virasoro algebra Virp,q with q = −1.
2 Some results on formal calculus
In this section, we shall review some formal variable notations and conventions, andwe formulate (and prove) certain basic results which we need in later sections.
First of all, throughout this paper all vector spaces are assumed to be over C (thefield of complex numbers) and N denotes the set of nonnegative integers. Lettersx, y, z, xi, yi, zi with i = 0, 1, . . . will be mutually commuting independent formalvariables.
For a vector space U , U [[x, x−1]] denotes the space of formal infinite integer powerseries with coefficients in U , U [[x]] denotes the space of formal infinite nonnegativepower series, and U((x)) denotes the space of lower truncated formal Laurent series.In particular, we have a vector space C[[x, x−1]], an algebra C[[x]], and a field C((x)).
We denote by FC[[x]] the field of fractions of C[[x]], which is naturally isomorphicto C((x)). Similarly, denote by FC[[x1,x2]] the field of fractions of C[[x1, x2]]. We define
ιx1,x2 : FC[[x1,x2]] → C((x1))((x2)) (2.1)to be the canonical extension of the embedding of C[[x1, x2]] into C((x1))((x2)).Following [FLM] and [FHL], we use the following formal variable convention: Forn ∈ Z,
(x1 − x2)n = ιx1,x2 ((x1 − x2)n) =∑i≥0
(n
i
)(−1)ixn−i1 xi2. (2.2)
For A(x) =∑
n∈Z A(n)xn ∈ U [[x, x−1]] with U a vector space, we have
ezd
dxA(x) = A(x+ z) =∑n∈Z
A(n)(x+ z)n,
ez(xd
dx)A(x) = A(xez).
We also use the following formal delta functions:
δ(x) =∑n∈Z
xn,
x−11 δ(x2x1
)=∑n∈Z
x−n−11 xn2 ,
x−10 δ(x1 − x2x0
)=∑n∈Z
x−n−10 (x1 − x2)n =∑n∈Z
∑i≥0
(n
i
)(−1)ix−n−10 xn−i1 xi2.
4
For n ≥ 0, we have1
n!
(∂
∂x2
)nx−11 δ
(x2x1
)= (x1 − x2)−n−1 − (−x2 + x1)−n−1. (2.3)
An important property is the following substitution rule:
f(x1, x2)δ
(x2x1
)= f(x2, x2)δ
(x2x1
), (2.4)
where f(x1, x2) is any series such that f(x2, x2) exists, e.g., f(x1, x2) ∈ U((x1, x2)).We shall frequently use the following simple facts:
Lemma 2.1. Let λ ∈ C× and let k be a positive integer. Then
(x1 − λx2)k(x2
∂
∂x2
)jδ
(λx2x1
)= 0 (2.5)
for 0 ≤ j ≤ k − 1.Proof. Using another independent formal variable z, we have
(x1 − λx2)k∑j≥0
zj
j!
(x2
∂
∂x2
)jδ
(λx2x1
)
= (x1 − λx2)kezx2∂
∂x2 δ
(λx2x1
)
= (x1 − λx2)kδ(λx2e
z
x1
)
= (λx2)k(ez − 1)kδ
(λx2e
z
x1
)∈ zkC[[x±11 , x±12 , z]].
Then it follows immediately.
Lemma 2.2. Let λ1, . . . , λr be distinct nonzero complex numbers, let k be a non-negative integer, and let Aij(x) ∈ U [[x, x−1]] for 1 ≤ i ≤ r, 0 ≤ j ≤ k, where U is avector space. Then
r∑i=1
k∑j=0
Aij(x2)
(x2
∂
∂x2
)jδ
(λix2x1
)= 0 (2.6)
if and only if Aij(x) = 0 for all i, j.
5
Proof. We only need to prove the “only if” part. For 1 ≤ i ≤ r, set
pi(x) =(x− λ1)k+1 · · · (x− λr)k+1
(x− λi)k+1 ∈ C[x].
Then there exist q1(x), . . . , qr(x) ∈ C[x] such thatp1(x)q1(x) + · · ·+ pr(x)qr(x) = 1. (2.7)
Let 1 ≤ s ≤ r be arbitrarily fixed. For 1 ≤ i ≤ r with i �= s, as (x− λi)k+1 dividesps(x), by Lemma 2.1 we have
ps(x1/x2)
(x2
∂
∂x2
)jδ
(λix2x1
)= 0 for 0 ≤ j ≤ k.
Then multiplying both sides of (2.6) by ps(x1/x2)qs(x1/x2), we get
k∑j=0
Asj(x2)ps(x1/x2)qs(x1/x2)
(x2
∂
∂x2
)jδ
(λsx2x1
)= 0.
Noticing that for 1 ≤ n ≤ r with n �= s and for 0 ≤ j ≤ k,
pn(x1/x2)qn(x1/x2)
(x2
∂
∂x2
)jδ
(λsx2x1
)= 0,
then using (2.7) we obtain
k∑j=0
Asj(x2)
(x2
∂
∂x2
)jδ
(λsx2x1
)= 0,
which amounts to
k∑j=0
Asj(x2)
(x2
∂
∂x2
)jδ
(x2x1
)= 0. (2.8)
For any nonzero integer m, by extracting the coefficient of x−m1 we get
k∑j=0
mjAsj(x2) = 0.
Taking m = 1, 2, . . . , k+1 and solving the system of equations we obtain Asj(x) = 0for 0 ≤ j ≤ k, as desired.
Let W be a vector space. Set
E(W ) = Hom(W,W ((x))) ⊂ (EndW )[[x, x−1]]. (2.9)
6
Lemma 2.3. Let a(x), b(x) ∈ E(W ), K(x1, x2) ∈ Hom(W,W ((x2))((x1))),p(x) = (x− λ1)k1 · · · (x− λr)kr ∈ C[x],
where λ1, . . . , λr are distinct nonzero complex numbers and k1, . . . , kr are positiveintegers. Then
p(x1/x2) (a(x1)b(x2)−K(x1, x2)) = 0 (2.10)if and only if
a(x1)b(x2)−K(x1, x2) =r∑
i=1
ki−1∑j=0
Aij(x2)
(x2
∂
∂x2
)jδ
(λix2x1
)(2.11)
for some Aij(x) ∈ E(W ), which are uniquely determined.Proof. The uniqueness follows from Lemma 2.2, so it remains to prove the existence.From assumption we have
p(x1/x2)a(x1)b(x2) = p(x1/x2)K(x1, x2),
which implies
p(x1/x2)a(x1)b(x2), p(x1/x2)K(x1, x2) ∈ Hom(W,W ((x1, x2))).Thus
a(x1)b(x2) = ιx1,x2
(1
p(x1/x2)
)A(x1, x2),
K(x1, x2) = ιx2,x1
(1
p(x1/x2)
)A(x1, x2)
for some A(x1, x2) ∈ Hom(W,W ((x1, x2))). Then
a(x1)b(x2)−K(x1, x2) =(ιx1,x2
(1
p(x1/x2)
)− ιx2,x1
(1
p(x1/x2)
))A(x1, x2).
Write
1
p(x)=
r∑i=1
ki∑j=0
aij(x− λi)j
with aij ∈ C. We then havea(x1)b(x2)−K(x1, x2)
= A(x1, x2)
(ιx1,x2
(1
p(x1/x2)
)− ιx2,x1
(1
p(x1/x2)
))
= A(x1, x2)r∑
i=1
ki∑j=1
(aijx
j2
(x1 − λix2)j −aijx
j2
(−λix2 + x1)j)
=r∑
i=1
ki∑j=1
aij1
(j − 1)!λ1−ji x
j2A(x1, x2)
(∂
∂x2
)j−1x−11 δ
(λix2x1
). (2.12)
7
It follows from induction that for any nonnegative integer n and for any B(x1, x2) ∈Hom(W,W ((x1, x2))), we have
B(x1, x2)
(∂
∂x2
)nx−11 δ
(λix2x1
)=
n∑i=0
Bi(x2)
(∂
∂x2
)iδ
(λix2x1
)(2.13)
for some Bi(x) ∈ E(W ). On the other hand, for n ≥ 1 we have(∂
∂x2
)n= x−n2
(x2
∂
∂x2
)(x2
∂
∂x2− 1)· · ·(x2
∂
∂x2− (n− 1)
). (2.14)
Using all these we get
a(x1)b(x2)−K(x1, x2) =r∑
i=1
ki−1∑j=0
Aij(x2)
(x2
∂
∂x2
)jδ
(λix2x1
)
for some Aij(x) ∈ E(W ), as desired.We shall also need the following simple fact:
Lemma 2.4. Let W be a vector space, let A(x1, x2) ∈ Hom(W,W ((x1, x2))), andlet λ ∈ C×. If A(x1, x2) �= 0, then
A(x1, x2) = (x1 − λx2)kB(x1, x2)for some k ∈ N, B(x1, x2) ∈ Hom(W,W ((x1, x2))) with B(λx2, x2) �= 0.Proof. Notice that for any m ∈ Z,
xm1 − (λx2)m = (x1 − λx2)Fm(x1, x2)for some Fm(x1, x2) ∈ C[x±11 , x±12 ]. For any C(x1, x2) ∈ Hom(W,W ((x1, x2))), ifC(λx2, x2) = 0, then
C(x1, x2) = C(x1, x2)− C(λx2, x2) = (x1 − λx2)C̄(x1, x2)for some C̄(x1, x2) ∈ Hom(W,W ((x1, x2))). On the other hand, as A(x1, x2) �=0, A(x1, x2)w �= 0 for some w ∈ W , where A(x1, x2)w ∈ W ((x1, x2)). ThenA(x1, x2)w = x
r1x
s2G(x1, x2) for some r, s ∈ Z, G(x1, x2) ∈ W [[x1, x2]] (nonzero).
Assuming G(x1 + λx2, x2) =∑
m≥0Gm(x2)xm1 with Gm(x2) ∈ C[[x2]], we have
G(x1, x2) =∑m≥0
Gm(x2)(x1 − λx2)m.
It follows that G(x1, x2) = (x1 − λx2)nH(x1, x2) with n ∈ N and H(x1, x2) ∈W [[x1, x2]] such that H(λx2, x2) �= 0. Then
A(x1, x2)w = (x1 − λx2)nxr1xs2H(x1, x2).In view of these, there exists a largest nonnegative integer k such that A(x1, x2) =(x1 − λx2)kB(x1, x2) for some B(x1, x2) ∈ Hom(W,W ((x1, x2))) with B(λx2, x2) �=0.
8
3 Quantum vertex algebras and their modules
In this section, we recall the notions of (weak) quantum vertex algebras and theirmodules, including a conceptual construction.
We first recall the notion of weak quantum vertex algebra, which was formulatedand studied in [Li2] and [Li3].
Definition 3.1. A weak quantum vertex algebra is a vector space V equipped witha linear map
Y (·, x) : V → Hom(V, V ((x))) ⊂ (EndV )[[x, x−1]]v → Y (v, x) =
∑n∈Z
vnx−n−1 (where vn ∈ EndV ),
called the adjoint vertex operator map, and a vector 1 ∈ V , called the vacuum vector,satisfying the following conditions: For v ∈ V ,
Y (1, x)v = v, Y (v, x)1 ∈ V [[x]] and limx→0
Y (v, x)1 = v,
and for u, v ∈ V , there existu(i), v(i) ∈ V, fi(x) ∈ C((x)) for i = 1, . . . , r
such that
x−10 δ(x1 − x2x0
)Y (u, x1)Y (v, x2)
−x−10 δ(x2 − x1−x0
) r∑i=1
fi(x2 − x1)Y (v(i), x2)Y (u(i), x1)
= x−12 δ(x1 − x0x2
)Y (Y (u, x0)v, x2) (3.1)
(the S-Jacobi identity).The following was proved in [Li2]:
Proposition 3.2. In Definition 3.1, the S-Jacobi identity axiom can be equivalentlyreplaced by weak associativity: For u, v, w ∈ V , there exists a nonnegative integer lsuch that
(x0 + x2)lY (u, x0 + x2)Y (v, x2)w = (x0 + x2)
lY (Y (u, x0)v, x2)w, (3.2)
and S-locality: For any u, v ∈ V , there existu(i), v(i) ∈ V, fi(x) ∈ C((x)) for i = 1, . . . , r,
and a nonnegative integer k such that
(x1 − x2)kY (u, x1)Y (v, x2) = (x1 − x2)kr∑
i=1
fi(x2 − x1)Y (v(i), x2)Y (u(i), x1). (3.3)
9
Let V be a weak quantum vertex algebra. Define a linear operator D on V byD(v) = v−21 for v ∈ V . Then
[D, Y (v, x)] = Y (D(v), x) = ddxY (v, x) (3.4)
for v ∈ V . It was proved in [Li2] that the S-locality relation (3.3) amounts to
Y (u, x)v =r∑
i=1
fi(−x)exDY (v(i),−x)u(i) (3.5)
(the S-skew symmetry).Definition 3.3. Let V be a weak quantum vertex algebra. A V -module is a vectorspace W equipped with a linear map
YW (·, x) : V → Hom(W,W ((x))) ⊂ (EndW )[[x, x−1]],satisfying the conditions that YW (1, x) = 1W (the identity operator on W ) and thatfor any u, v ∈ V, w ∈ W , there exists a nonnegative integer l such that
(x0 + x2)lYW (u, x0 + x2)YW (v, x2)w = (x0 + x2)
lYW (Y (u, x0)v, x2)w. (3.6)
We have (see [Li2]):
Proposition 3.4. Let V be a weak quantum vertex algebra and let (W,YW ) be aV -module. Then
YW (Dv, x) = ddxYW (v, x) for v ∈ V. (3.7)
For u, v ∈ V , let u(i), v(i) ∈ V, fi(x) ∈ C((x)) (i = 1, . . . , r) such that S-Jacobiidentity (3.1) holds. Then
x−10 δ(x1 − x2x0
)YW (u, x1)YW (v, x2)
−x−10 δ(x2 − x1−x0
) r∑i=1
fi(x2 − x1)YW (v(i), x2)YW (u(i), x1)
= x−12 δ(x1 − x0x2
)YW (Y (u, x0)v, x2). (3.8)
We next recall from [Li2] the conceptual construction of weak quantum vertexalgebras and modules.
Definition 3.5. Let W be a vector space and let a(x), b(x) ∈ E(W ). The orderedpair (a(x), b(x)) is said to be compatible if there exists k ∈ N such that
(x1 − x2)ka(x1)b(x2) ∈ Hom(W,W ((x1, x2))). (3.9)
10
Let (a(x), b(x)) be a compatible pair in E(W ). Define a(x)nb(x) ∈ E(W ) forn ∈ Z in terms of generating function
YE(a(x), z)b(x) =∑n∈Z
a(x)nb(x)z−n−1
by
YE(a(x), z)b(x) = z−k((x1 − x)ka(x1)b(x)
) |x1=x+z, (3.10)where k is any nonnegative integer such that (3.9) holds.
Definition 3.6. A subset U of E(W ) is said to be S-local if for any a(x), b(x) ∈ U ,there exist
a(i)(x), b(i)(x) ∈ U, fi(x) ∈ C((x)) (i = 1, . . . , r)such that
(x1 − x2)ka(x1)b(x2) = (x1 − x2)kr∑
i=1
fi(x2 − x1)b(i)(x2)a(i)(x1) (3.11)
for some nonnegative integer k.
Note that if U is an S-local subset of E(W ), then every pair (a(x), b(x)) in U iscompatible. The following result was obtained in [Li2]:
Theorem 3.7. Every S-local subset U of E(W ) generates a weak quantum vertexalgebra 〈U〉 with W as a canonical module with YW (α(x), z) = α(z) for α(x) ∈ 〈U〉.
Let U be a vector space. Recall that a rational quantum Yang-Baxter operatoron U is a linear map
S(x) : U ⊗ U → U ⊗ U ⊗ C((x)),satisfying
S12(x)S13(x+ z)S23(z) = S23(z)S13(x+ z)S12(x) (3.12)(the quantum Yang-Baxter equation), where for 1 ≤ i < j ≤ 3,
S ij(x) : U ⊗ U ⊗ U → U ⊗ U ⊗ U ⊗ C((x))denotes the canonical extension of S(x). It is said to be unitary if
S(x)S21(−x) = 1,where S21(x) = σS(x)σ with σ denoting the flip operator on U ⊗ U .
For a weak quantum vertex algebra V , following [EK] let
Y (x) : V ⊗ V → V ((x)) (3.13)be the canonical linear map associated to the vertex operator map Y (·, x).
The following notion, which was formulated in [Li2], was derived from Etingof-Kazhdan’s notion of quantum vertex operator algebra (see [EK]):
11
Definition 3.8. A quantum vertex algebra is a weak quantum vertex algebra Vequipped with a unitary rational quantum Yang-Baxter operator S(x) on V , satis-fying the following conditions:
S(x)(1⊗ v) = 1⊗ v for v ∈ V, (3.14)[D ⊗ 1,S(x)] = − d
dxS(x), (3.15)
Y (u, x)v = exDY (−x)S(−x)(v ⊗ u) for u, v ∈ V, (3.16)S(x1)(Y (x2)⊗ 1) = (Y (x2)⊗ 1)S23(x1)S13(x1 + x2), (3.17)
where D is the linear operator on V defined by D(v) = v−21 for v ∈ V . Wesometimes denote a quantum vertex algebra by a pair (V,S).
The following notion was due to Etingof-Kazhdan (see [EK]):
Definition 3.9. Let V be a weak quantum vertex algebra. For any positive integern, define a linear map
Zn : V⊗n ⊗ C((x1))((x2)) · · · ((xn))→ V ((x1))((x2)) · · · ((xn))
byZn(v
(1) ⊗ · · · ⊗ v(n) ⊗ f) = fY (v(1), x1) · · ·Y (v(n), xn)1for v(1), . . . , v(n) ∈ V, f ∈ C((x1))((x2)) · · · ((xn)). V is said to be non-degenerate iffor every positive integer n, Zn is injective.
The following result can be found in [Li2] (cf. [EK]):
Proposition 3.10. Let V be a weak quantum vertex algebra. Assume that V isnon-degenerate. Then there exists a linear map S(x) : V ⊗ V → V ⊗ V ⊗ C((x)),which is uniquely determined by
Y (u, x)v = exDY (−x)S(−x)(v ⊗ u) for u, v ∈ V.
Furthermore, (V,S) carries the structure of a quantum vertex algebra and the fol-lowing relation holds
[1⊗D,S(x)] = ddxS(x). (3.18)
Remark 3.11. Note that a quantum vertex algebra was defined as a pair (V,S).In view of Proposition 3.10, the term “a non-degenerate quantum vertex algebra”without reference to a quantum Yang-Baxter operator is unambiguous. If a weakquantum vertex algebra V is of countable dimension over C and if V as a V -moduleis irreducible, then V is non-degenerate by a result of [Li3] (Corollary 3.10). In viewof this, the term “irreducible quantum vertex algebra” is also unambiguous.
12
We conclude this section with some basic notions. The notions of homomor-phism, isomorphism and automorphism for weak quantum vertex algebras are de-fined in the obvious way. For example, an automorphism of a weak quantum vertexalgebra V is a bijective linear endomorphism σ of V such that σ(1) = 1 and
σ(Y (u, x)v) = Y (σ(u), x)σ(v) for u, v ∈ V.
Definition 3.12. A subset U of a weak quantum vertex algebra V is called agenerating subset if V is linearly spanned by vectors
u(1)n1 · · ·u(r)nr 1
for r ∈ N, u(i) ∈ U, ni ∈ Z (1 ≤ i ≤ r).
4 φ-coordinated quasi modules for weak quantum
vertex algebras
In this section, we first recall from [Li6] the definition of a φ-coordinated quasimodule for a weak quantum vertex algebra and the conceptual construction. Wethen define the notions of T -type weak quantum vertex algebra and G-equivariantφ-coordinated quasi module. As one of the main results we establish a commutatorformula.
We begin with a convention. Note that for any p(x) ∈ C[x], p(ex) ∈ C[[x]].One can show (cf. [Li6]) that p(ex) = 0 if and only if p(x) = 0. Recall thatC(x) denotes the field of rational functions. For any g(x) = p(x)/q(x) ∈ C(x) withp(x), q(x) ∈ C[x], we define
g(ex) =p(ex)
q(ex)∈ FC[[x]] = C((x)). (4.1)
Definition 4.1. Let V be a weak quantum vertex algebra. A φ-coordinated quasiV -module is a vector space W equipped with a linear map
YW (·, x) : V → Hom(W,W ((x))) ⊂ (EndW )[[x, x−1]],
satisfying the conditions that YW (1, x) = 1W and that for u, v ∈ V , there exists anonzero polynomial p(x) such that
p(x1/x2)YW (u, x1)YW (v, x2) ∈ Hom(W,W ((x1, x2))) (4.2)
and
p(ez)YW (Y (u, z)v, x2) = (p(x1/x2)YW (u, x1)YW (v, x2)) |x1=x2ez . (4.3)
13
Remark 4.2. The parameter φ in Definition 4.1 refers to the formal series φ(x, z) =xez, which is a particular associate of the one-dimensional additive formal group(law) F (x, y) = x+ y, as defined in [Li6].
The following was proved in [Li6] (Proposition 5.6):
Proposition 4.3. Let V be a weak quantum vertex algebra, let u, v ∈ V , and let(W,YW ) be a φ-coordinated quasi V -module. Suppose that
(x1 − x2)kY (u, x1)Y (v, x2)
= (x1 − x2)kr∑
i=1
ιx2,x1(fi(ex1−x2))Y (v(i), x2)Y (u(i), x1), (4.4)
where k ∈ N, fi(x) ∈ C(x), u(i), v(i) ∈ V , and suppose that p(x) is any nonzeropolynomial such that
p(x1/x2)YW (u, x1)YW (v, x2) ∈ Hom(W,W ((x1, x2))).Then
p(x1/x2)YW (u, x1)YW (v, x2)
= p(x1/x2)r∑
i=1
ιx2,x1 (fi(x1/x2))YW (v(i), x2)YW (u
(i), x1). (4.5)
Set
log(1 + z) =∑n≥1
(−1)n−1 zn
n∈ zC[[z]]. (4.6)
Notice that∑
n≥1(−1)n−1 zn−1n
is a unit in C[[z]]. For m ∈ Z, it is understood that
(log(1 + z))m = zm
(∑n≥1
(−1)n−1 zn−1
n
)m∈ zmC[[z]] ⊂ C((z)).
The following was proved in [Li6] (Lemma 5.8)2:
Lemma 4.4. Let W be a vector space and let
A(x1, x2) ∈ Hom(W,W ((x1))((x2))), B(x1, x2) ∈ Hom(W,W ((x2))((x1))),C(x0, x2) ∈ Hom(W,W ((x2))((x0))).
If there exists a nonnegative integer k such that
(x1 − x2)kA(x1, x2) = (x1 − x2)kB(x1, x2),((x1 − x2)kA(x1, x2)
) |x1=x2ex0 = xk2(ex0 − 1)kC(x0, x2),2There is a typo in [Li6]: B(x2, x1) in (5.18) and in the proof should be B(x1, x2).
14
then
(zx2)−1δ
(x1 − x2zx2
)A(x1, x2)− (zx2)−1δ
(x2 − x1−zx2
)B(x1, x2)
= x−11 δ(x2(1 + z)
x1
)C(log(1 + z), x2). (4.7)
Furthermore, the converse is also true.
The following result generalizes Proposition 5.9 of [Li6]:
Proposition 4.5. Let V be a weak quantum vertex algebra and let (W,YW ) be aφ-coordinated quasi V -module. Let u, v ∈ V . Assume
u(i), v(i) ∈ V, fi(x) ∈ C(x) (i = 1, . . . , r)
such that (4.4) holds for some nonnegative integer k. Then there exists a nonzeropolynomial p(x) such that
(xz)−1δ(x1 − xxz
)p(x1/x)YW (u, x1)YW (v, x)
−(xz)−1δ(x− x1−xz
)p(x1/x)
r∑i=1
ιx1,x (fi(x1/x))YW (v(i), x)YW (u
(i), x1)
= x−11 δ(x(1 + z)
x1
)p(x1/x)YW (Y (u, log(1 + z))v, x) . (4.8)
Proof. From definition, there exists a nonzero polynomial p(x) such that
p(x1/x2)YW (u, x1)YW (v, x2) ∈ Hom(W,W ((x1, x2))) (4.9)
and
p(ez)YW (Y (u, z)v, x2) = (p(x1/x2)YW (u, x1)YW (v, x2)) |x1=x2ez . (4.10)
With (4.9), by Proposition 4.3 we have
p(x1/x2)YW (u, x1)YW (v, x2)
= p(x1/x2)r∑
i=1
ιx2,x1 (fi(x1/x2))YW (v(i), x2)YW (u
(i), x1). (4.11)
Then it follows immediately from Lemma 4.4 (with k = 0).
Motivated by Proposition 4.3 we formulate the following notion:
15
Definition 4.6. A weak quantum vertex algebra V is said to be of T-type if for anyu, v ∈ V , there exist
u(i), v(i) ∈ V, fi(x) ∈ C(x) (i = 1, . . . , r)
such that (4.4) holds for some nonnegative integer k.
Recall that a trigonometric quantum Yang-Baxter operator on a vector space Uis a linear map
S(x) : U ⊗ U → U ⊗ U ⊗ C(x),satisfying quantum Yang-Baxter equation
S12(x)S13(xz)S23(z) = S23(z)S13(xz)S12(x). (4.12)
It is said to be unitary if S21(1/x)S(x) = 1.Definition 4.7. A quantum vertex algebra (V,S(x)) is said to be of T-type if thereexists a unitary trigonometric quantum Yang-Baxter operator T (x) on V such thatS(x) = T (ex).
The following technical result follows from the proof of Proposition 2.6 in [LTW]with C((x)) replaced by C(x) in a few places:
Lemma 4.8. Let V be a weak quantum vertex algebra. Suppose that U is a gen-erating subset of V such that the condition in Definition 4.6 with U in place of Vholds. Then V is of T-type.
Next, we recall from [Li6] the conceptual construction of weak quantum vertexalgebras and their φ-coordinated quasi modules. Let W be a vector space whichis fixed for a while. Let a(x), b(x) ∈ E(W ). Assume that there exists a nonzeropolynomial p(x) such that
p(x/z)a(x)b(z) ∈ Hom(W,W ((x, z))). (4.13)
Define a(x)enb(x) ∈ E(W ) for n ∈ Z in terms of generating function
Y eE (a(x), z)b(x) =∑n∈Z
(a(x)enb(x))z−n−1
by
Y eE (a(x), z)b(x) = p(ez)−1 (p(x1/x)a(x1)b(x)) |x1=xez , (4.14)
where p(x) is any nonzero polynomial such that (4.13) holds and p(ez)−1 denotesthe inverse of p(ez) in C((z)).
16
Definition 4.9. A subset U of E(W ) is said to be quasi Strig-local if for anya(x), b(x) ∈ U , there exist
a(i)(x), b(i)(x) ∈ U, fi(x) ∈ C(x) (i = 1, . . . , r)
such that
p(x/z)a(x)b(z) = p(x/z)r∑
i=1
ιz,x(f(z/x))b(i)(z)a(i)(x) (4.15)
for some nonzero polynomial p(x). A subset U of E(W ) is quasi Strig-compatible iffor any a(x), b(x) ∈ U , there exists a nonzero polynomial p(x) such that
p(x/z)a(x)b(z) ∈ Hom(W,W ((x, z))).
Note that relation (4.15) implies (4.13), so that every quasi Strig-local subset isautomatically quasi Strig-compatible.
The following strengthens one of the main results in [Li6]:
Theorem 4.10. Let W be a vector space and let U be any quasi Strig-local subset ofE(W ). Then U generates a weak quantum vertex algebra 〈U〉e with W as a faithful φ-coordinated quasi module where YW (α(x), z) = α(z) for α(x) ∈ 〈U〉e. Furthermore,〈U〉e is of T-type.Proof. The first part was proved in [Li6] (Theorem 5.4). Then we have a weakquantum vertex algebra 〈U〉e with U as a generating subset. Let a(x), b(x) ∈ U .There exist
a(i)(x), b(i)(x) ∈ U, fi(x) ∈ C(x) (i = 1, . . . , r)such that
p(x/z)a(x)b(z) = p(x/z)r∑
i=1
ιz,x(f(z/x))b(i)(z)a(i)(x)
for some nonzero polynomial p(x). By Proposition 5.3 of [Li6], we have
(x1 − x2)kY eE (a(x), x1)Y eE (b(x), x2)
= (x1 − x2)kr∑
i=1
ιx2,x1(fi(ex2−x1))Y eE (b
(i)(x), x2)YeE (a
(i)(x), x1)
for some nonnegative integer k. Then it follows from Lemma 4.8 that 〈U〉e is ofT-type.
We have the following result (cf. [Li6], Proposition 4.11):
17
Proposition 4.11. Let V be a T-type weak quantum vertex algebra and let (W,YW )be a φ-coordinated quasi V -module. Set VW = {YW (v, x) | v ∈ V }. Then VW is aquasi Strig-local subspace of E(W ), (VW , Y eE , 1W ) is a weak quantum vertex algebra,and YW is a homomorphism of weak quantum vertex algebras.
Proof. Combining Definition 4.1 with Proposition 4.3, we see that VW is a quasiStrig-local subspace of E(W ). Let u, v ∈ V . By definition, there exists a nonzeropolynomial p(x) such that
p(x1/x)YW (u, x1)YW (v, x) ∈ Hom(W,W ((x1, x)))and
p(ez)YW (Y (u, z)v, x) = (p(x1/x)YW (u, x1)YW (v, x)) |x1=xez .On the other hand, from the definition of Y eE (·, x) we have
p(ez)Y eE (YW (u, x), z)YW (v, x) = (p(x1/x)YW (u, x1)YW (v, x)) |x1=xez .Consequently,
p(ez)Y eE (YW (u, x), z)YW (v, x) = p(ez)YW (Y (u, z)v, x).
Since both Y eE (YW (u, x), z)YW (v, x) and YW (Y (u, z)v, x) involve only finitely manynegative powers of z, by cancellation we obtain
Y eE (YW (u, x), z)YW (v, x) = YW (Y (u, z)v, x).
It follows that (VW , YeE , 1W ) is a weak quantum vertex algebra and YW is a homo-
morphism of weak quantum vertex algebras.
Note that a(λx) ∈ E(W ) for any λ ∈ C×, a(x) ∈ E(W ). For λ ∈ C×, defineRλ ∈ End(E(W )) by
Rλ(a(x)) = a(λx) for a(x) ∈ E(W ). (4.16)This gives rise to a group action of C× on E(W ). From now on, we fix this particularaction.
Proposition 4.12. Let W be a vector space, let Γ be a subgroup of C×, and let U be aquasi Strig-local subset of E(W ) such that U is Γ-stable. Then Γ is an automorphismgroup of the weak quantum vertex algebra 〈U〉e generated by U .Proof. Let a(x), b(x) ∈ 〈U〉e. Then there exists a nonzero polynomial p(x) such that
p(x1/x)a(x1)b(x) ∈ Hom(W,W ((x1, x))).For any nonzero complex number λ we have
p(x1/x)a(λx1)b(λx) ∈ Hom(W,W ((x1, x))).
18
Then we get
Y eE (a(λx), z)b(λx)
= p(ez)−1 (p(x1/x)a(λx1)b(λx)) |x1=xez=
[p(ez)−1 (p(x̄1/x̄)a(x̄1)b(x̄)) |x̄1=x̄ez
] |x̄=λx= [Y eE (a(x̄), z)b(x̄)]x̄=λx .
It follows from induction that 〈U〉e is Γ-stable and Γ acts on weak quantum vertexalgebra 〈U〉e by automorphisms.Lemma 4.13. Let W be a vector space, let a(x), b(x) ∈ E(W ), and let
K(x1, x2) ∈ Hom(W,W ((x2))((x1))).
Assume
a(x1)b(x2)−K(x1, x2) =r∑
j=0
Aj(x2)1
j!
(x2
∂
∂x2
)jδ
(x2x1
)
+k∑
i=1
s∑j=0
Bij(x2)1
j!
(x2
∂
∂x2
)jδ
(λix2x1
), (4.17)
where Aj(x), Bij(x) ∈ E(W ), λi ∈ C× with λi �= 1 for 1 ≤ i ≤ k. Then
a(x)ejb(x) = Aj(x) for 0 ≤ j ≤ r,a(x)ejb(x) = 0 for j > r. (4.18)
Proof. Set
q(x) =k∏
i=1
((x− 1)r+1 − (λi − 1)r+1
)s+1 ∈ C[x].Noticing that
q(x) = q̄(x)k∏
i=1
(x− λi)s+1
for some q̄(x) ∈ C[x], we have
q(x1/x2)
(x2
∂
∂x2
)jδ
(λix2x1
)= 0
for 0 ≤ j ≤ s, 1 ≤ i ≤ k. Then we get
q(x1/x2) (a(x1)b(x2)−K(x1, x2)) =r∑
j=0
q(x1/x2)Aj(x2)1
j!
(x2
∂
∂x2
)jδ
(x2x1
).(4.19)
19
This implies
(x1/x2 − 1)r+1q(x1/x2)(a(x1)b(x2)−K(x1, x2)) = 0.
From definition we have
(ex0 − 1)r+1q(ex0)Y eE (a(x), x0)b(x) =((x1/x− 1)r+1q(x1/x)a(x1)b(x)
) |x1=xex0 .With the above two identities, by Lemma 4.4 we obtain
x−11 δ(x(1 + z)
x1
)q(x1/x)Y
eE (a(x), log(1 + z))b(x)
= (xz)−1δ(x1 − xxz
)q(x1/x)a(x1)b(x)
−(xz)−1δ(x− x1−xz
)q(x1/x)K(x1, x).
Applying Resz to both sides, we get
q(x1/x)(a(x1)b(x)−K(x1, x))= Resx0q(x1/x)Y
eE (a(x), x0)b(x)e
x0x∂
∂x δ
(x
x1
)
=∑i≥0
a(x)ei b(x)1
i!q(x1/x)
(x∂
∂x
)iδ
(x
x1
), (4.20)
which is a finite sum. Combining this with (4.19) we obtain
∑i≥0
(Ai(x)− a(x)ib(x)) 1i!q(x1/x)
(x∂
∂x
)iδ
(x
x1
)= 0, (4.21)
where we set Ai(x) = 0 for i > r. Write
q(x1/x) = (x1/x− 1)r+1P (x1/x) + α,
where P (x) ∈ C[x] and
α = (−1)k(s+1)k∏
i=1
(λi − 1)(r+1)(s+1).
Note that α �= 0 as λi �= 1 for 1 ≤ i ≤ k. Since
(x1/x− 1)r+1(x∂
∂x
)jδ
(x
x1
)= 0 for 0 ≤ j ≤ r,
20
(4.21) reduces to
∑i≥0
(Ai(x)− a(x)ib(x)) 1i!α
(x∂
∂x
)iδ
(x
x1
)= 0, (4.22)
which impliesα (Ai(x)− a(x)ib(x)) = 0 for i ≥ 0.
Then our assertions follow immediately.
Using Lemma 4.13 we have the following generalization:
Proposition 4.14. Let W be a vector space, let a(x), b(x) ∈ E(W ), and letK(x1, x2) ∈ Hom(W,W ((x2))((x1))).
Assume
a(x1)b(x2)−K(x1, x2) =k∑
i=1
r∑j=0
Aij(x2)1
j!
(x2
∂
∂x2
)jδ
(λix2x1
), (4.23)
where Aij(x) ∈ E(W ) and λi ∈ C× distinct for 1 ≤ i ≤ k. Thena(λix)
ejb(x) = Aij(x) for 0 ≤ j ≤ r,
a(λix)ejb(x) = 0 for j > r. (4.24)
Proof. For λ ∈ C×, from (4.23) we have
a(λx1)b(x2)−K(λx1, x2) =k∑
i=1
r∑j=0
Aij(x2)1
j!
(x2
∂
∂x2
)jδ
(λix2λx1
).
Taking λ = λi with 1 ≤ i ≤ k, using Lemma 4.13 we geta(λix)
ejb(x) = Aij(x) for 0 ≤ j ≤ r
and a(λix)ejb(x) = 0 for j > r, as desired.
Combining Lemma 2.3 with Proposition 4.14 we immediately have:
Corollary 4.15. Let W be a vector space, let a(x), b(x) ∈ E(W ), and letK(x1, x2) ∈ Hom(W,W ((x2))((x1))).
Assume
p(x1/x2)a(x1)b(x2) = p(x1/x2)K(x1, x2), (4.25)
where p(x) = (x−λ1)k1 · · · (x−λr)kr with λ1, . . . , λr distinct nonzero complex num-bers and with ki ≥ 1. Then
a(x1)b(x)−K(x1, x) = Resx0r∑
i=1
Y eE (a(λix), x0)b(x)ex0(x ∂∂x)δ
(λix
x1
). (4.26)
21
In the following we shall establish certain technical results.
Lemma 4.16. Let a(x), b(x) ∈ E(W ) and let λ ∈ C×. Suppose that k is an integerand p(x) is a polynomial with p(λ) �= 0 such that
(x1/x2 − λ)kp(x1/x2)a(x1)b(x2) ∈ Hom(W,W ((x1, x2))).Then
a(λx)enb(x) = 0 for all n ≥ k. (4.27)Proof. From assumption we have
(x1/x2 − 1)kp(λx1/x2)a(λx1)b(x2) ∈ Hom(W,W ((x1, x2))).Then
zkY eE (a(λx), z)b(x)
= zk(ez − 1)−kp(λez)−1 ((x1/x− 1)kp(λx1/x)a(λx1)b(x)) |x1=xez ,where p(λez)−1 denotes the inverse of p(λez) in C((z)). We have zk(ez−1)−k ∈ C[[z]]and p(λez)−1 ∈ C[[z]] as p(λ) �= 0. We know that(
(x1/x− 1)kp(λx1/x)a(λx1)b(x)) |x1=xez
contains only nonnegative powers of z. Then it follows that zkY eE (a(λx), z)b(x)involves only nonnegative powers of z. Thus, a(λx)enb(x) = 0 for all n ≥ k.Lemma 4.17. Let a(x), b(x) ∈ E(W ). Suppose that p(x) is a nonzero polynomialsuch that
p(x1/x2)a(x1)b(x2) ∈ Hom(W,W ((x1, x2))).Write p(x) = (x − 1)sq(x) with s ∈ N, q(x) ∈ C[x] such that q(1) �= 0. Then fork ∈ Z,
(x1/x2 − 1)kq(x1/x2)a(x1)b(x2) ∈ Hom(W,W ((x1, x2))) (4.28)if and only if a(x)ejb(x) = 0 for j ≥ k.Proof. The “only if” part follows from Lemma 4.16. Now, assume that a(x)ejb(x) = 0for j ≥ k. We have
a(x1)b(x) = ιx1,x(1/p(x1/x))A(x1, x)
for some A(x1, x) ∈ Hom(W,W ((x1, x))). If A(x1, x) = 0, there is nothing to prove.Assume A(x1, x) �= 0. By Lemma 2.4 we have A(x1, x) = (x1/x − 1)rB(x1, x) forsome r ∈ N, B(x1, x) ∈ Hom(W,W ((x1, x))) with B(x, x) �= 0. Then
(x1/x− 1)s−rq(x1/x)a(x1)b(x) = B(x1, x) ∈ Hom(W,W ((x1, x))), (4.29)
22
so that
Y eE (a(x), z)b(x) = (ez − 1)r−sq(ez)−1B(xez, x).
We have
limz→0
zs−rY eE (a(x), z)b(x) = limz→0
(ez − 1z
)r−sq(ez)−1B(xez, x) = q(1)−1B(x, x).
We get a(x)enb(x) = 0 for n ≥ s − r and a(x)es−r−1b(x) �= 0. As we are given thata(x)emb(x) for m ≥ k, we must have k ≥ s − r. Then it follows from (4.29) that(4.28) holds.
Definition 4.18. Let V be a T-type weak quantum vertex algebra and let G be anautomorphism group equipped with a linear character χ : G→ C×. A G-equivariantφ-coordinated quasi V -module is a φ-coordinated quasi V -module (W,YW ), satisfyingthe conditions that
YW (gv, x) = YW (v, χ(g)x) for g ∈ G, v ∈ V, (4.30)and that for u, v ∈ V , there exists p(x) ∈ C[x] with only zeroes in χ(G) such that
p(x1/x2)YW (u, x1)YW (v, x2) ∈ Hom(W,W ((x1, x2))). (4.31)As the main result of this section we have:
Theorem 4.19. Let V,G, χ be given as in Definition 4.18 and let (W,YW ) be aG-equivariant φ-coordinated quasi V -module. Suppose that u, v ∈ V ,
u(i), v(i) ∈ V, fi(x) ∈ C(x) (i = 1, . . . , r)such that
(x1 − x2)kY (u, x1)Y (v, x2) = (x1 − x2)kr∑
i=1
ιx2,x1fi(ex1−x2)Y (v(i), x2)Y (u(i), x1)
for some nonnegative integer k. Then there are finitely many g1, . . . , gn ∈ G withχ(g1), . . . , χ(gn) distinct such that
YW (u, x1)YW (v, x)−r∑
i=1
ιx1,x (fi(x1/x))YW (v(i), x)YW (u
(i), x1)
= Resx0
n∑j=1
YW (Y (gj(u), x0)v, x)ex0(x ∂∂x)δ
(χ(gj)x
x1
). (4.32)
Furthermore, if the linear character χ : G→ C× is injective, we have
YW (u, x1)YW (v, x)−r∑
i=1
ιx1,x (fi(x1/x))YW (v(i), x)YW (u
(i), x1)
= Resx0∑g∈G
YW (Y (gu, x0)v, x)ex0(x ∂∂x)δ
(χ(g)x
x1
), (4.33)
which is a finite sum.
23
Proof. From definition, there are distinct nonzero complex numbers λ1, . . . , λn ∈χ(G) such that
p(x1/x2)YW (u, x1)YW (v, x2) ∈ Hom(W,W ((x1, x2))),where p(x) = (x− λ1)k1 · · · (x− λn)kn with ki ≥ 1. By Proposition 4.3 we have
p(x1/x2)YW (u, x1)YW (v, x2)
= p(x1/x2)r∑
i=1
ιx1,x2 (fi(x1/x2))YW (v(i), x2)YW (u
(i), x1).
Then by Corollary 4.15 we have
YW (u, x1)YW (v, x)−r∑
i=1
ιx1,x (fi(x1/x))YW (v(i), x)YW (u
(i), x1)
= Resx0
n∑j=1
Y eE(YW (u, λjx), x0
)YW (v, x)e
x0(x ∂∂x)δ
(λjx
x1
). (4.34)
For 1 ≤ j ≤ n, let gj ∈ G be such that λj = χ(gj). ThenYW (u, λjx) = YW (u, χ(gj)x) = YW (gju, x).
Using the fact that YW (·, x) is a homomorphism (from Proposition 4.11), we have
Y eE(YW (u, λjx), x0
)YW (v, x) = Y
eE(YW (gju, x), x0
)YW (v, x) = YW (Y (gju, x0)v, x).
Then we obtain
YW (u, x1)YW (v, x)−r∑
i=1
ιx1,x (fi(x1/x))YW (v(i), x)YW (u
(i), x1)
= Resx0
n∑j=1
YW (Y (gj(u), x0)v, x)ex0(x ∂∂x)δ
(λjx
x1
),
proving the first assertion.Assume that χ is injective. For g ∈ G, if g �= gj for 1 ≤ j ≤ n, with χ(g) �= λj
we have p(χ(g)) �= 0. By Lemma 4.16 we get YW (u, χ(g)x)emYW (v, x) = 0 for m ≥ 0.Then
YW (Y (gu, x0)v, x) = YeE (YW (gu, x), x0)YW (v, x) = Y
eE (YW (u, χ(g)x), x0)YW (v, x),
involving only nonnegative powers of x0. Thus
Resx0YW (Y (gu, x0)v, x)ex0(x ∂∂x)δ
(χ(g)x
x1
)= 0.
Then the second assertion follows.
24
As an immediate consequence of Theorem 4.19 we have:
Corollary 4.20. Let V be a T-type weak quantum vertex algebra and let (W,YW )be a G-equivariant φ-coordinated quasi V -module. Let u, v ∈ V . Then
p(x1/x2)YW (u, x1)YW (v, x2) ∈ Hom(W,W ((x1, x2))), (4.35)
where p(x) is a product of (x− χ(σ))kσ with σ ∈ G and kσ ≥ 1 such that
(σu)kσ−1v �= 0 and (σu)nv = 0 for n ≥ kσ.
The following is a useful technical result:
Lemma 4.21. Let V be a T -type weak quantum vertex algebra, let G be an auto-morphism group with a linear character χ, and let (W,YW ) be a φ-coordinated quasiV -module. Suppose that U is a generating subset of V such that
YW (gu, x) = YW (u, χ(g)x) for u ∈ U, g ∈ G (4.36)
and such that {YW (u, x) | u ∈ U} is χ(G)-quasi Strig-local. Then (W,YW ) is aG-equivariant φ-coordinated quasi V -module.
Proof. Set K = {v ∈ V | YW (gv, x) = YW (v, χ(g)x) for g ∈ G}. From assumption,we have U ∪ {1} ⊂ K. Let u, v ∈ K, g ∈ G. As YW (·, x) is a homomorphism ofweak quantum vertex algebras by Proposition 4.11, we get
YW (gY (u, z)v, x) = YW (Y (gu, z)gv, x)
= Y eE (YW (gu, x), z)YW (gv, x)
= Y eE (YW (u, χ(g)x), z)YW (v, χ(g)x)
= YW (Y (u, z)v, χ(g)x)
for g ∈ G. This shows unv ∈ K for n ∈ Z. Since U generates V , we have V = K,proving that (4.36) holds with V in place of U , i.e., (4.30) holds.
Recall Lemma 5.2 from [Li6] that every quasi Strig-local subset of E(W ) isquasi compatible. Let Γ be a subgroup of C×. We say that a finite sequenceψ1(x), . . . , ψr(x) in E(W ) is Γ-quasi compatible if there exists a nonzero polynomialp(x) with only zeros in Γ such that( ∏
1≤i
On the other hand, recall Proposition 4.9 in [Li6]: Let
ψ1(x), . . . , ψr(x), a(x), b(x), φ1(x), . . . , φs(x) ∈ E(W ).Assume that the ordered sequences (a(x), b(x)) and
(ψ1(x), . . . , ψr(x), a(x), b(x), φ1(x), . . . , φs(x))
are quasi compatible. Then for any n ∈ Z, the ordered sequence(ψ1(x), . . . , ψr(x), a(x)
enb(x), φ1(x), . . . , φs(x))
is quasi compatible. The same proof with the obvious changes shows that the refine-ment of Proposition 4.9 with quasi-compatibility replaced by Γ-quasi compatibilityholds. It then follows from induction that for any Γ-quasi compatible subset A ofE(W ), 〈A〉e is Γ-quasi compatible.
Note that YW (·, x) is a homomorphism of weak quantum vertex algebras fromV to VW where VW = {YW (v, x) | v ∈ V }. Since U is a generating subset of V ,it follows that VW = 〈UW 〉e with UW = {YW (u, x) | u ∈ U}. Then VW is χ(G)-quasi compatible. In particular, (4.31) holds. Therefore, (W,YW ) is a G-equivariantφ-coordinated quasi V -module.
As we need, recall the following result from [Li6] (Lemma 6.7):
Lemma 4.22. Let W be a vector space and let a(x), b(x) ∈ E(W ). Assume thatthere exist
0 �= p(x) ∈ C[x], qi(x) ∈ C((x)), u(i)(x), v(i)(x) ∈ E(W ) (1 ≤ i ≤ r)such that
p(x1/x2)a(x1)b(x2) =r∑
i=1
qi(x1/x2)u(i)(x2)v
(i)(x1). (4.38)
Then (a(x), b(x)) is quasi compatible and
p(ex0)Y eE (a(x), x0)b(x)
= Resx1
(1
x1 − xex0 p(x1/x)a(x1)b(x)−1
−xex0 + x1r∑
i=1
qi(x1/x)u(i)(x)v(i)(x1)
).
Furthermore, if k is the order of zero of p(x) at 1, then a(x)enb(x) = 0 for n ≥ k and1
k!p(k)(1)a(x)ek−1b(x)
= Resx1
(1
x1 − xp(x1/x)a(x1)b(x)−1
−x+ x1 q(x1/x)u(i)(x)v(i)(x1)
). (4.39)
26
5 Deformed Virasoro algebra Virp,q with q = −1In this section, we study the deformed Virasoro algebra Virp,q with q = −1 in thecontext of quantum vertex algebras and φ-coordinated quasi modules. We presenta canonical connection of this algebra with a Clifford vertex super-algebra and itsG-equivariant φ-coordinated quasi modules.
We first recall from [SKAO] the definition of the deformed Virasoro algebraVirp,q. Let p and q be nonzero complex numbers with p not a root of −1, so thatpn + 1 �= 0 for every integer n. Set
t =q
p.
Definition 5.1. The deformed Virasoro algebra Virp,q is defined to be the associa-tive unital algebra over C with generators Tn (n ∈ Z), subject to relations
∞∑l=0
fl(Tm−lTn+l − Tn−lTm+l) = −(1− q)(1− p/q)1− p (p
m − p−m)δm+n,0 (5.1)
for m,n ∈ Z, where the coefficients fl’s are given by
f(z) ≡∞∑l=0
flzl = exp
( ∞∑n=1
(1− qn)(1− t−n)1 + pn
zn
n
). (5.2)
Remark 5.2. Note that in general, fl �= 0 for infinitely many l, so that the expres-sion on the left-hand side of (5.1) is a genuine infinite sum. Because of this, thealgebra Virp,q is a topological algebra (involving a formal completion). This can bedone rigorously by imitating Frenkel-Zhu’s construction of the universal envelopingalgebra U(V ) of a vertex operator algebra V (see [FZ]).
Form a generating function
T (x) =∑n∈Z
Tnx−n. (5.3)
Then the defining relations (5.1) can be written as
f(z/x)T (x)T (z)− f(x/z)T (z)T (x)= −(1− q)(1− p/q)
1− p[δ(pzx
)− δ
(z
px
)]. (5.4)
From the defining relations (5.1), it can be readily seen that Virp,q is a Z-gradedalgebra with deg Tn = −n for n ∈ Z. Then one can study highest weight Virp,q-modules, in a way similar to that for the ordinary Virasoro algebra, and in fact, thehighest weight Virp,q-modules have been studied in [SKAO] and [BP]. One can alsodefine restricted modules, which are more general than highest weight modules.
27
The deformed Virasoro algebra Virp,q with q = −1 has been extensively studiedin [BP] from an associative algebra point of view. In this case, one has
f(z) =1 + z
1− z = 1 + 2∞∑l=1
zl.
The defining relations for Virp,−1 read as
[Tm, Tn] + 2∑l≥0
(Tm−lTn+l − Tn−lTm+l) = −2(1 + p
1− p)(pm − p−m)δm+n,0 (5.5)
for m,n ∈ Z. In terms of the generating function T (x) =∑n∈Z Tnx−n, we have(1 + x2
x1
1− x2x1
)T (x1)T (x2)−
(1 + x1
x2
1− x1x2
)T (x2)T (x1)
= −2(1 + p
1− p)(
δ
(px2x1
)− δ
(p−1x2x1
)). (5.6)
We say a Virp,−1-module W is restricted if for any w ∈ W , Tnw = 0 for nsufficiently large, namely, T (x) ∈ E(W ).
Throughout this section, we assume that p is not a root of unity. Set
Γp = {pn | n ∈ Z} ⊂ C×. (5.7)Next, we introduce a vertex super-algebra.
Definition 5.3. Let E be a complex vector space with a basis {e(r) | r ∈ Z} andequip E with a bilinear form 〈·, ·〉 defined by
〈e(r), e(s)〉 = 2(δr,s+1 + δr,s−1) for r, s ∈ Z. (5.8)It can be readily seen that this form is symmetric and non-degenerate. Set
L(E) = E ⊗ C[t, t−1].Let � be a complex number. Define a bilinear form 〈·, ·〉� on L(E) by
〈e(r) ⊗ tm, e(s) ⊗ tn〉� = 2�(δr,s+1 + δr,s−1)δm+n+1,0 (5.9)for r, s,m, n ∈ Z. This form is symmetric and non-degenerate whenever � �= 0.Denote by Cl(L(E), �) the associated Clifford algebra, which by definition is theassociative unital algebra generated by vector space L(E), subject to relations
(a⊗ tm)(b⊗ tn) + (b⊗ tn)(a⊗ tm) = 〈a, b〉δm+n+1,0� (5.10)for a, b ∈ E, m, n ∈ Z.
28
Remark 5.4. The Clifford algebra Cl(L(E), �) can be defined alternatively as thequotient algebra of the universal enveloping algebra of the Lie super-algebra
Ê = (E ⊗ C[t, t−1])⊕ Ck,modulo relation k = �, where Ê0 = Ck, Ê1 = E ⊗ C[t, t−1], and [k, Ê] = 0,
[a⊗ tm, b⊗ tn] = 〈a, b〉δm+n+1,0kfor a, b ∈ E, m, n ∈ Z.
Following the tradition, for a ∈ E, n ∈ Z we alternatively denote a ⊗ tn by an.For a ∈ E, set
a(x) =∑n∈Z
anx−n−1 ∈ Cl(L(E), �)[[x, x−1]]. (5.11)
Set
VÊ(�, 0) = Cl(L(E), �)/Cl(L(E), �)(E ⊗ C[t]), (5.12)a left Cl(L(E), �)-module, and set
1 = 1 + Cl(L(E), �)(E ⊗ C[t]) ∈ VÊ(�, 0).Identify E as a subspace of VÊ(�, 0) through the map a → a(−1)1. It is well known(see [FFR]) that there exists a vertex super-algebra structure on VÊ(�, 0), which isuniquely determined by the condition that 1 is the vacuum vector and Y (a, x) = a(x)for a ∈ E.
As we need later, we mention some simple facts about VÊ(�, 0). The subspace Egenerates VÊ(�, 0) as a vertex super-algebra. Furthermore, for r, s ∈ Z, we have
e(r)n e(s) = δn,0�〈e(r), e(s)〉1 = 2�δn,0(δr,s+1 + δr,s−1)1 for n ≥ 0. (5.13)
We also have
e(r)−1e
(r) = 0 for r ∈ Z. (5.14)Lemma 5.5. Assume that � is a nonzero complex number. Then VÊ(�, 0) viewed asa Cl(L(E), �)-module is irreducible and VÊ(�, 0) viewed as a vertex super-algebra issimple.
Proof. Let k be any positive integer. Set E[k] = span{e(r) | − k ≤ r ≤ k+1} (evendimensional). One can show that the bilinear form of E, restricted to E[k], is non-degenerate. Let V [k] denote the vertex super subalgebra of VÊ(�, 0), generated byE[k]. From [FFR], V [k] is an irreducible Cl(L(E[k]), �)-module. It is clear that V [k]with k ≥ 1 form an increasing filtration of VÊ(�, 0). Then it follows that VÊ(�, 0) isan irreducible Cl(L(E), �)-module. This particularly implies that VÊ(�, 0) is a simplevertex super-algebra.
29
For the rest of this section we consider � = 1. The following is straightforward:
Lemma 5.6. For m ∈ Z, there exists an automorphism σm of VÊ(1, 0), which isuniquely determined by
σm(e(r)) = e(m+r) for r ∈ Z.
Furthermore, this gives rise to a group action of Z on VÊ(1, 0) by automorphisms.
By identifying m with σm for m ∈ Z, we view Z as an automorphism group ofVÊ(1, 0). Define a linear character of Z:
χp : Z → C× with χp(n) = pn for n ∈ Z. (5.15)Now, we are in a position to present our first main result of this section.
Theorem 5.7. Let W be any restricted Virp,−1-module. Then there exists a Z-equivariant φ-coordinated quasi VÊ(1, 0)-module structure YW (·, x) which is uniquelydetermined by the condition
YW (e(r), x) = T (prx) for r ∈ Z.
Proof. As E generates VÊ(1, 0) as a vertex super-algebra, the uniqueness is clear.We now prove the existence. Set
UW = {T (pmx) | m ∈ Z} ⊂ E(W ).From (5.6) we have(
x1 + ps−rx2
x1 − ps−rx2
)T (prx1)T (p
sx2)−(x2 + p
r−sx1x2 − pr−sx1
)T (psx2)T (p
rx1)
= −2(1 + p
1− p)(
δ
(ps+1−rx2
x1
)− δ
(ps−1−rx2
x1
))(5.16)
for r, s ∈ Z. By multiplying both sides of (5.16) by a suitable polynomial we get(x1/x2 − ps+1−r)(x1/x2 − ps−1−r)(x1/x2 + ps−r)T (prx1)T (psx2)
= −(x1/x2 − ps+1−r)(x1/x2 − ps−1−r)(x1/x2 + ps−r)T (psx2)T (prx1). (5.17)From definition, UW is Γp-stable and from (5.17), UW is quasi Strig-local. In viewof Theorem 4.10, UW generates a weak quantum vertex algebra 〈UW 〉e with W as afaithful φ-coordinated quasi module.
We next show that 〈UW 〉e becomes a Cl(L(E), 1)-module by letting e(r)(z) actas Y eE (T (p
rx), z) for r ∈ Z. To this end, we need to proveY eE (T (p
rx), x1)YeE (T (p
sx), x2) + YeE (T (p
sx), x2)YeE (T (p
rx), x1)
= 2(δr,s+1 + δs,r+1)x−11 δ
(x2x1
)(5.18)
30
for r, s ∈ Z. With (5.17), by Proposition 5.3 of [Li6], we haveP (ex1−x2)Y eE (T (p
rx), x1)YeE (T (p
sx), x2) = −P (ex1−x2)Y eE (T (psx), x2)Y eE (T (prx), x1),where P (x) = (x−ps+1−r)(x−ps−1−r)(x+ps−r). As p is not a root of−1, ex1−x2+ps−ris invertible in C[[x1, x2]]. By cancellation we get
(ex1−x2 − ps+1−r)(ex1−x2 − ps−1−r)Y eE (T (prx), x1)Y eE (T (psx), x2)= −(ex1−x2 − ps+1−r)(ex1−x2 − ps−1−r)Y eE (T (psx), x2)Y eE (T (prx), x1). (5.19)
If r−s �= ±1, (ex1−x2−ps+1−r) and (ex1−x2−ps−1−r) are also invertible in C[[x1, x2]],so that we furthermore obtain
Y eE (T (prx), x1)Y
eE (T (p
sx), x2) = −Y eE (T (psx), x2)Y eE (T (prx), x1). (5.20)This proves that (5.18) holds for r �= s± 1.
Consider the case r = s± 1. Set h(x) =∑n≥1 1n!xn−1. Then ex − 1 = xh(x). Ifr = s+ 1, we have
(ex1−x2 − ps+1−r)(ex1−x2 − ps−1−r) = (x1 − x2)h(x1 − x2)(ex1−x2 − p−2),where h(x1 − x2)(ex1−x2 − p−2) is invertible in C[[x1, x2]]. From (5.19) we get
(x1 − x2)Y eE (T (prx), x1)Y eE (T (psx), x2)= −(x1 − x2)Y eE (T (psx), x2)Y eE (T (prx), x1). (5.21)
If r = s − 1, similarly we obtain the same relation. As 〈UW 〉e as a weak quantumvertex algebra is generated by T (pmx) for m ∈ Z, it follows that 〈U〉e is actually avertex super-algebra.
Recall that P (x) = (x − ps+1−r)(x − ps−1−r)(x + ps−r). For r = s + 1, we haveP (x) = (x− 1)(x− p−2)(x + p−1) and P ′(1) = (1− p−2)(1 + p−1). With (5.17), byLemma 4.22 we have
T (prx)enT (psx) = 0 for n ≥ 1
and
P ′(1)T (prx)e0T (psx)
= Resx1
(1
x1 − xP (x1/x)T (prx1)T (p
sx) +1
−x+ x1P (x1/x)T (psx)T (prx1)
)= Resx1x
−1(x1/x− p−2)(x1/x− p−1) ··((
x1 + p−1x
x1 − p−1x)T (prx1)T (p
sx)−(p−1x+ x1p−1x− x1
)T (psx)T (prx1)
)= Resx1x
−1(x1/x− p−2)(x1/x− p−1) ··(−2)
(1 + p
1− p)(
δ
(x
x1
)− δ
(p−2xx1
))
= (1− p−2)(1− p−1)(−2)(1 + p
1− p)
= 2(1− p−2)(1 + p−1)(δr,s+1 + δs,r+1),
31
which impliesT (prx)e0T (p
sx) = 2 · 1W .It follows from the super-commutator formula that (5.18) holds for r = s + 1.Similarly, one can show that (5.18) holds for r = s−1. Thus 〈UW 〉e is a Cl(L(E), 1)-module with e(r)(z) acting as Y eE (T (p
rx), z) for r ∈ Z. Furthermore, we have
e(r)n 1W = T (prx)en1W = 0 for r ∈ Z, n ∈ N.
It follows from the construction of VÊ(1, 0) that there exists a Cl(L(E), 1)-modulehomomorphism ψ from VÊ(1, 0) to 〈UW 〉e, sending 1 to 1W . That is,
ψ(Y (e(r), z)v) = Y eE (T (prx), z)ψ(v) for r ∈ Z, v ∈ VÊ(1, 0).
Since E generates VÊ(1, 0) as a vertex super-algebra, it follows that ψ is a ho-momorphism of vertex super-algebras. As W is a φ-coordinated quasi module for〈U〉e, W becomes a φ-coordinated quasi VÊ(1, 0)-module through homomorphismψ. Furthermore, we have
YW (σn(e(r)), x) = YW (e
(r+n), x) = T (pr+nx) = YW (e(r), pnx) = YW (e
(r), χp(n)x)
for n, r ∈ Z. As E generates VÊ(1, 0), by Lemma 4.21, it is also Z-equivariant.Therefore, W is a Z-equivariant φ-coordinated quasi VÊ(1, 0)-module.
On the other hand, we have:
Theorem 5.8. Let (W,YW ) be a Z-equivariant φ-coordinated quasi VÊ(1, 0)-module.Then W is a restricted module for Virp,−1 with T (x) = YW (e(1), x).Proof. As YW (e
(1), x) ∈ E(W ) from definition, we must prove(x1 + x2x1 − x2
)YW (e
(1), x1)YW (e(1), x2)−
(x2 + x1x2 − x1
)YW (e
(1), x2)YW (e(1), x1)
= −2(1 + p
1− p)(
δ
(px2x1
)− δ
(p−1x2x1
)). (5.22)
Note that for n ∈ Z, i ≥ 0, we have
(σne(1))ie
(1) = e(n+1)i e
(1) = δi,0〈e(n+1), e(1)〉1 = 2δi,0(δn,1 + δn,−1)1.
In view of Theorem 4.19, we have
YW (e(1), x1)YW (e
(1), x2) + YW (e(1), x2)YW (e
(1), x1)
= 2
(δ
(px2x1
)+ δ
(p−1x2x1
)), (5.23)
32
noticing that χp is one-to-one. From this we obtain
(x1 − px2)(px1 − x2)YW (e(1), x1)YW (e(1), x2)= −(x1 − px2)(px1 − x2)YW (e(1), x2)YW (e(1), x1),
which implies
(x1 − px2)(px1 − x2)YW (e(1), x1)YW (e(1), x2) ∈ Hom(W,W ((x1, x2))). (5.24)Note that e
(1)j e
(1) = 0 for j ≥ −1. Using the fact that YW is a homomorphism(from Proposition 4.11), we get
YW (e(1), x)ejYW (e
(1), x) = YW (e(1)j e
(1), x) = 0 for j ≥ −1.From Lemma 4.17 we have
(x1 − x2)−1(x1 − px2)(px1 − x2)YW (e(1), x1)YW (e(1), x2) ∈ Hom(W,W ((x1, x2))).Thus
(x1 − px2)(px1 − x2)YW (e(1), x1)YW (e(1), x2) = (x1 − x2)A(x1, x2),(x1 − px2)(px1 − x2)YW (e(1), x2)YW (e(1), x1) = (x2 − x1)A(x1, x2) (5.25)
for some A(x1, x2) ∈ Hom(W,W ((x1, x2))). ThenYW (e
(1), x1)YW (e(1), x2) + YW (e
(1), x2)YW (e(1), x1)
=
(x1 − x2
(x1 − px2)(px1 − x2) +x2 − x1
(x2 − px1)(px2 − x1))A(x1, x2)
=1
p+ 1
(x−11 δ
(px2x1
)+ x−12 δ
(px1x2
))A(x1, x2). (5.26)
Comparing this with (5.23) we get
x−11 δ(px2x1
)A(x1, x2) = 2(p+ 1)δ
(px2x1
),
x−12 δ(px1x2
)A(x1, x2) = 2(p+ 1)δ
(px1x2
).
Using these identities we obtain(x1 + x2x1 − x2
)YW (e
(1), x1)YW (e(1), x2)−
(x2 + x1x2 − x1
)YW (e
(1), x2)YW (e(1), x1)
=
(x1 + x2
(x1 − px2)(px1 − x2) −x1 + x2
(px2 − x1)(x2 − px1))A(x1, x2)
=1
p− 1(
1
x1 − px2 −1
px1 − x2 +1
px2 − x1 −1
x2 − px1
)A(x1, x2)
=1
p− 1(x−11 δ
(px2x1
)− x−12 δ
(px1x2
))A(x1, x2)
= −2(1 + p
1− p)(
δ
(px2x1
)− δ
(p−1x2x1
)).
33
Therefore, W is a restricted module for Virp,−1 with T (x) = YW (e(1), x).
References
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